Noise-to-State Stability of Random Coupled Kuramoto Oscillators via Feedback Control

Abstract

This paper is intended to study noise-to-state stability in probability (NSSP) for random coupled Kuramoto oscillators with input control (RCKOIC). A feedback control is designed, which makes us give the existence and uniqueness of a solution for RCKOIC. Based on Kirchhoff’s matrix tree theorem in graph theory, an original and appropriate Lyapunov function for RCKOIC is established. With the help of the Lyapunov method and by resorting to some analysis skills, NSSP for RCKOIC with an arbitrarily coupled topological structure and second-order moment process stochastic disturbance is acquired. Finally, the effectiveness of the obtained results is verified by a numerical test and its simulation process.

1. Introduction

The coupled Kuramoto oscillators (CKO) model proposed by Y. Kuramoto [1] is one of the most representative models for studying complex dynamical systems, which has been widely used in various fields due to its practicability, such as biology [2,3], chemistry [4,5], physics [6,7], and engineering [8]. However, in plenty of actual situations, CKO is always inevitably disturbed by external noise, which is simulated by many scholars using white noise as a diffusion form. Nevertheless, white noise is not well suited for the simulation of CKO in some physical environments, since it is not differentiable almost everywhere in theory, and the ideal white noise does not exist in practice because of its infinite bandwidth. From a practical standpoint, we typically use a stationary random process to replicate the external noise interference affecting CKO. Thus, general noise disturbances such as bounded noise and second-order moment processes are fully considered for CKO, which is ordinarily called random coupled Kuramoto oscillators (RCKO).

On the other hand, the practical application of RCKO mainly depends on its dynamic behavior. Therefore, in the past few decades, analyzing and controlling the dynamic behavior of RCKO to meet the expected requirements has beeb one of the hot issues in the scientific community. In order to achieve the desired stability in practical applications, feedback control methods have become one of the common means of controlling such stochastic coupled systems [9,10]. Based on the above deliberation, we examine the RCKO system with input control (RCKOIC) as described below:

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}{\theta }_i\left(t\right)}{\mathrm{d}t}}={\omega }_i-\sum _{j=1}^n{a}_{ij}sin\left({\theta }_j\left(t\right)-{\theta }_i\left(t\right)\right)+u_i\left(t\right)+{\sigma }_{i}cos{\theta }_i\left(t\right){\xi }_i\left(t\right),i=1,2,\cdots ,n,\end{array}$$

(1)

where ${\theta }_i\in {\mathbb{R}}^1$ denotes the phase of the i-th oscillator, and ${\omega }_i$ refers to its natural frequency. Matrix $A={\left(a_{ij}\right)}_{n\times n}(a_{ij}\ge 0)$ represents the coupling configuration of RCKOIC (1), and $a_{ij}$ stands for the coupling strength. Input control is represented by $u_i\left(t\right):{\mathbb{R}}^{+}\to {\mathbb{R}}^1$. The disturbance intensity is denoted by ${\sigma }_{i}cos{\theta }_i$, and ${\xi }_i\left(t\right)\in {\mathbb{R}}^1$ is a second-order moment stochastic process that is ${\mathcal{F}}_t$-adapted.

Noise-to-state stability is an extension from deterministic systems to stochastic systems, compared with the well-known input-to-state stability. However, as far as the authors know, there are few results to study the noise-to-state stability of RCKOIC (1) due to the following reasons: RCKOIC (1) has no theoretical framework results, such as the existence and uniqueness of solution theorem. And RCKOIC (1) is nonlinear, so it has very rich dynamic behavior. In addition, the dynamic behavior of RCKOIC (1) depends on not only each oscillator but also the coupling topological structure between them. And feedback control can be an effective means to help the system cope with random perturbations and maintain the stability and desired dynamic behavior of the system. Building on the aforementioned discussion, the primary objective of this article is to investigate noise-to-state stability in probability (NSSP) for RCKOIC (1).

The contributions of this paper can be summarized in the following three key aspects. Firstly, through a feedback control strategy designed by us, the existence and uniqueness of a solution for RCKOIC (1) is provided. Secondly, a novel Lyapunov function for RCKOIC (1) is developed utilizing Kirchhoff’s matrix tree theorem from graph theory. By combining the Lyapunov method with novel analytical techniques, we achieve NSSP for RCKOIC (1) under arbitrary coupled topological structures and second-order moment stochastic disturbances. Thirdly, a numerical example along with simulation results is provided to confirm the validity and applicability of our theoretical findings.

The rest of this paper is organized as follows: Section 2 provides some preliminaries and model description. The main results are presented in Section 3, followed by numerical simulations in Section 4 to demonstrate the effectiveness of our work. Lastly, the conclusion is provided in Section 5.

2. Preliminaries and Model Description

The following notations are used throughout the paper. ${\mathbb{R}}^n$ is the n-dimensional real vector space, ${\mathbb{R}}^{+}=[0,+\infty )$, and ${\mathbb{R}}^{m\times n}$ denotes the set of $m\times n$-dimensional real matrices. The superscript “T” stands for the transpose of the matrix or vector. Let $\left|x\right|={\left({\sum }_{i=1}^n{x}_i^2\right)}^{\frac{1}{2}}$ and $\parallel A\parallel ={\left({\sum }_{i=1}^m{\sum }_{j=1}^n{a}_{ij}^2\right)}^{\frac{1}{2}}$ denote the norm of vector $x={(x_1,x_2,\cdots ,x_n)}^{\mathrm{T}}\in {\mathbb{R}}^n$ and matrix $A={\left(a_{ij}\right)}_{m\times n}\in {\mathbb{R}}^{m\times n}$, respectively.

The probability space $\left(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P}\right)$ is considered, where $\mathbb{E}$ denotes the expectation under $\mathbb{P}$, ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ denotes a filtration meeting the standard requirements, and $\mathbb{P}$ is the probability measure. Assume that a stochastic process vector defined on this entire probability space is $\xi \left(t\right)\in {\mathbb{R}}^n$. We define the set ${\mathcal{K}}_{\infty }$ as $\{\alpha (·)\mid \alpha (·):{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}\mathrm{such}\mathrm{that}\alpha (·)\mathrm{is}\mathrm{strictly}\mathrm{increasing},\mathrm{unbounded},\mathrm{and}\alpha \left(0\right)=0\}$. Likewise, ${\mathcal{K}}_{\mathcal{L}}$ is defined as $\{\beta (·,·)\mid \beta (·,·):{\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+}\mathrm{where}\beta (·,y)\in {\mathcal{K}}_{\infty }\mathrm{for}\mathrm{every}\mathrm{fixed}y\ge 0$ and $\beta (x,·)\mathrm{is}\mathrm{strictly}\mathrm{decreasing}\mathrm{to}0\mathrm{as}y\to +\infty \mathrm{for}\mathrm{any}\mathrm{fixed}x\ge 0\}$. Definitions for other symbols will be provided as they are introduced.

A directed graph is widely recognized as a useful tool for describing RCKOIC (1). By utilizing a weighted directed graph $(\mathcal{G},A)$ with n nodes $(n\ge 2)$, where $\mathcal{G}$ consists of a vertex set and a directed edge set, and the weight matrix $A={\left(a_{ij}\right)}_{n\times n}$ represents the coupling structure of RCKOIC (1), we can effectively and clearly represent the dynamics of RCKOIC (1). Each vertex represents an oscillator in RCKOIC (1), and directed arcs between vertices depict the interactions between oscillators. Especially if $a_{ij}=0$, there is no interaction between the two oscillators. In addition, the related content of graph theory is more seen in [11,12]. Moreover, to make the readers more convenient, we write RCKOIC (1) in a more concise format as follows:

$${\displaystyle \frac{\mathrm{d}\theta \left(t\right)}{\mathrm{d}t}}=F\left(\theta \left(t\right),t\right)+G\left(\theta \left(t\right),t\right)\xi \left(t\right),$$

where $\theta ={({\theta }_1,\cdots ,{\theta }_n)}^{\mathrm{T}}\in {\mathbb{R}}^n,$ the stochastic process $\xi ={({\xi }_1,\cdots ,{\xi }_n)}^{\mathrm{T}}\in {\mathbb{R}}^n$ is ${\mathcal{F}}_t$-adapted satisfying ${\displaystyle \underset{0\le s\le t}{sup}}\mathbb{E}{\left|\xi \left(s\right)\right|}^2<+\infty$,

$$F\left(\theta ,t\right)={\left(\begin{array}{c}{\omega }_1-{\displaystyle \sum _{j=1}^n}{a}_{1j}sin({\theta }_j-{\theta }_1)+u_1,\cdots ,{\omega }_n-{\displaystyle \sum _{j=1}^n}{a}_{nj}sin({\theta }_j-{\theta }_n)+u_n\end{array}\right)}^{\mathrm{T}}\in {\mathbb{R}}^n,$$

and

$$G\left(\theta ,t\right)=\left(\begin{array}{cccc}{\sigma }_{1}cos{\theta }_1& 0& \cdots & 0\\ 0& {\sigma }_{2}cos{\theta }_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\sigma }_{n}cos{\theta }_n\end{array}\right)\in {\mathbb{R}}^{n\times n}.$$

At the end of this section, let us give two useful definitions and a lemma to close this part.

Definition 1. 

The solution of RCKOIC (1) is a stochastic process, which is defined as $\theta \left(t\right)=\theta (t;{\theta }_0,0)$, where ${\theta }_0$ is the initial phase, satisfying the following two conditions:

1 

$\theta \left(t\right)$ is almost surely continuous with respect to t and ${\mathcal{F}}_t$-adapted, where $t\in \left[0,T\right]$, $\left(T<+\infty \right)$.

2 

$\theta \left(t\right)={\theta }_0+{\int }_0^{t}F\left(\theta ,s\right)\mathrm{d}s+{\int }_0^{t}G\left(\theta ,s\right)\xi \left(s\right)\mathrm{d}sa.s.$

Definition 2. 

Consider any $\epsilon >0$, $t\in {\mathbb{R}}^{+}$, and an initial condition ${\theta }_0\in {\mathbb{R}}^n$. If there exist functions $\beta (·,·)\in {\mathcal{K}}_{\mathcal{L}}$ and $\Gamma (·)\in {\mathcal{K}}_{\infty }$ such that

$$\begin{array}{c}\hfill \mathbb{P}\left\{\left|\theta \left(t\right)\right|\le \beta \left(\right|{\theta }_0|,t)+\Gamma \left(\underset{0\le s\le t}{sup}\mathbb{E}{\left|\xi \left(s\right)\right|}^2\right)\right\}\ge 1-\epsilon ,\end{array}$$

then, the RCKOIC $\left(1\right)$ is said to exhibit noise-to-state stability in probability (NSSP).

Lemma 1 

(Kirchhoff’s Matrix Tree Theorem [11]). Let $n\ge 2$ define $c_i$ as the cofactor of the i-th diagonal element in the Laplacian matrix of a weighted digraph $(\mathcal{G},A)$. Then, we have

$$\begin{array}{c}\hfill c_i=\sum _{\mathcal{T}\in {\mathbb{T}}_i}W\left(\mathcal{T}\right),i=1,2,\cdots ,n,\end{array}$$

where ${\mathbb{T}}_i$ denotes the collection of all spanning trees $\mathcal{T}$ in the weighted digraph $(\mathcal{G},A)$ rooted at the vertex i, and $W\left(\mathcal{T}\right)$ represents the weight associated with $\mathcal{T}$. In particular, if $(\mathcal{G},A)$ is a strongly connected digraph, then $c_i>0$.

3. Main Results

In this section, we present a theorem ensuring that RCKOIC (1) satisfies NSSP as outlined below.

Theorem 1. 

We design feedback control $u_i=-{\rho }_i{\theta }_i-{\omega }_i$$(i=1,\cdots ,n),$ where the gain coefficient

$$\begin{array}{c}\hfill {\rho }_i>{\displaystyle \frac{1}{2}}\left(\sum _{j=1}^{n}4a_{ij}+{\sigma }_i\right),\end{array}$$

(2)

then, RCKOIC (1) is NSSP.

Proof. 

The proof of Theorem 1 is comparatively skillful. Thus, for the convenience of readers, we divide the proof into two parts.

Step 1: Above all, we manifest the solution of RCKOIC (1) is existent and unique for any $t\in {\mathbb{R}}^{+}$.

(Existence). Together with feedback control $u_i$ designed above, by calculation, it yeilds

$$\begin{array}{ccc}& & |F(\theta ,t)-F(\overline{\theta },t)|+||G(\theta ,t)-G(\overline{\theta },t)\left|\right|\hfill \\ & \le & \sum _{i,j=1}^n{a}_{ij}\left(\right|{\overline{\theta }}_j-{\theta }_j|+|{\overline{\theta }}_i-{\theta }_i\left|\right)+\sum _{i=1}^n{\rho }_i|{\overline{\theta }}_i-{\theta }_i|+{\left[\sum _{i=1}^n{\sigma }_i^2{(cos{\overline{\theta }}_i-cos{\theta }_i)}^2\right]}^{\frac{1}{2}}\hfill \\ & \le & \left[\sum _{i,j=1}^n(2a_{ij}+{\rho }_i)+{\left(\sum _{i=1}^n{\sigma }_i^2\right)}^{\frac{1}{2}}\right]|\overline{\theta }-\theta |\triangleq L|\overline{\theta }-\theta |,\hfill \end{array}$$

where $L={\displaystyle \sum _{i,j=1}^n}(2a_{ij}+{\rho }_i)+{\left({\displaystyle \sum _{i=1}^n}{\sigma }_i^2\right)}^{\frac{1}{2}}$. Then, we construct an iterative sequence as follows:

$${\theta }^{\left(k+1\right)}\left(t\right)={\theta }_0+{\int }_0^{t}F\left({\theta }^{\left(k\right)},s\right)\mathrm{d}s+{\int }_0^{t}G\left({\theta }^{\left(k\right)},s\right)\xi \left(s\right)\mathrm{d}s,k=0,1,2,\cdots ,$$

where ${\theta }^{\left(0\right)}\left(t\right)={\theta }_0$ represents the initial value. In terms of mathematical induction, one can derive

$$\begin{array}{ccc}\hfill |{\theta }^{\left(k\right)}\left(t\right)-{\theta }^{\left(k-1\right)}\left(t\right)|& \le & {\int }_0^t|F\left({\theta }^{\left(k-1\right)},s\right)-F\left({\theta }^{\left(k-2\right)},s\right)|\mathrm{d}s\hfill \\ & & +{\int }_0^t\left|\right|G\left({\theta }^{\left(k-1\right)},s\right)-G\left({\theta }^{\left(k-2\right)},s\right)\left|\right|·|\xi \left(s\right)|\mathrm{d}s\hfill \\ & \le & |{\theta }_0|{\displaystyle \frac{{\left(L{\int }_0^t\left(1+|\xi \left(s\right)|\right)\mathrm{d}s\right)}^k}{k!}}.\hfill \end{array}$$

In accordance with the discriminant method of optimal series, apparently we can know that ${\sum }_{k=1}^{\infty }|{\theta }^{\left(k\right)}\left(t\right)-{\theta }^{\left(k-1\right)}\left(t\right)|$ is uniform convergence. Obviously, when $k\to \infty$, the limit of ${\theta }^{\left(k\right)}\left(t\right)$ exists, which is represented by $\theta \left(t\right)$. Therefore, we can obtain

$$\theta \left(t\right)={\theta }_0+{\int }_0^{t}F\left(\theta ,s\right)\mathrm{d}s+{\int }_0^{t}G\left(\theta ,s\right)\xi \left(s\right)\mathrm{d}sa.s.$$

Evidently, ${\theta }_0$ is ${\mathcal{F}}_t$-adapted so that ${\theta }_1$ is ${\mathcal{F}}_t$-adapted. After iteration, it is easy to see that ${\theta }_k$ is ${\mathcal{F}}_t$-adapted, so that $\theta \left(t\right)$ is ${\mathcal{F}}_t$-adapted, and it is almost surely continuous for any $t\in \left[0,T\right]$. Combining with Definition 2, the solution of RCKOIC (1) exists.

(Uniqueness). We assume that $\theta \left(t\right)$ and $\overline{\theta }\left(t\right)$ are both the solutions of RCKOIC (1). Let ${\epsilon }_q>0$ and $\underset{q\to \infty }{lim}{\epsilon }_q=0$. On grounds of ${\displaystyle \underset{0\le s\le t}{sup}}\mathbb{E}{\left|\xi \left(s\right)\right|}^2<+\infty$, obviously we can know that $\left|\xi \right(s\left)\right|<+\infty$ a.s. Then, based on Bellman–Gronwall Lemma [13], let $q\to +\infty$, we can see

$$\begin{array}{ccc}\hfill |\theta \left(t\right)-\overline{\theta }\left(t\right)|& \le & {\int }_0^t|F(\theta ,s)-F(\overline{\theta },s)|\mathrm{d}s+{\int }_0^t\left|\right|G(\theta ,s)-G(\overline{\theta },s)\left|\right|·|\xi \left(s\right)|\mathrm{d}s\hfill \\ & \le & {\epsilon }_q+L{\int }_0^t|\theta -\overline{\theta }\left|\right(1+\left|\xi \left(s\right)\right|)\mathrm{d}s\hfill \\ & \le & {\epsilon }_q{e}^{L{\int }_0^T(1+|\xi \left(s\right)\left|\right)\mathrm{d}s}=0,\hfill \end{array}$$

which means the solution of RCKOIC (1) is unique.

Step 2: Soon afterwards, we show that RCKOIC (1) is NSSP.

It is clear that RCKOIC (1) is equivalent to the following system:

$$\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}{\theta }_i\left(t\right)}{\mathrm{d}t}}& =& {\omega }_i-{\displaystyle \sum _{j=1}^n}{b}_{ij}sin\left({\theta }_j\left(t\right)-{\theta }_i\left(t\right)\right)+u_i\left(t\right)+{\sigma }_{i}cos{\theta }_i\left(t\right){\xi }_i\left(t\right)\hfill \\ & & +{\displaystyle \sum _{j=1}^n}{\tau }_{ij}sin\left({\theta }_j\left(t\right)-{\theta }_i\left(t\right)\right),i=1,2,\cdots ,n,\hfill \end{array}$$

(3)

where $b_{ij}=a_{ij}+{\tau }_{ij}>0$, and sufficiently small positive real number ${\tau }_{ij}$ satisfies ${\sum }_{j=1}^n{\tau }_{ij}<{\displaystyle \frac{1}{10}}\left(2{\rho }_i-{\sum }_{j=1}^{n}4a_{ij}-{\sigma }_i\right)$. Then, we give the truncated system of system (3) as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}{\theta }_i\left(t\right)}{\mathrm{d}t}}={\omega }_i-\sum _{j=1}^n{b}_{ij}sin\left({\theta }_j\left(t\right)-{\theta }_i\left(t\right)\right)+u_i\left(t\right)+{\sigma }_{i}cos{\theta }_i\left(t\right){\xi }_i\left(t\right),i=1,2,\cdots ,n.\end{array}$$

(4)

Now, we consider a Lyapunov function

$$\begin{array}{c}\hfill V(\theta ,t)=\sum _{i=1}^n{e}^{\eta t}{c}_i{\theta }_i^2,\end{array}$$

where the positive constant $\eta <{\displaystyle \frac{1}{10}}\left(2{\rho }_i-{\sum }_{j=1}^{n}4a_{ij}-{\sigma }_i\right)$, and $c_i$ denotes the cofactor of the i-th diagonal entry in the Laplacian matrix.

$$C=\left(\begin{array}{cccc}{\displaystyle \sum _{k\ne 1}}{b}_{1k}& -b_{12}& \cdots & -b_{1s}\\ -b_{21}& {\displaystyle \sum _{k\ne 2}}{b}_{2k}& \cdots & -b_{2s}\\ ⋮& ⋮& \ddots & ⋮\\ -b_{s1}& -b_{s2}& \cdots & {\displaystyle \sum _{k\ne s}}{b}_{sk}\end{array}\right)$$

of the weighted digraph $(\mathcal{G},B)$, and weight matrix $B={\left(b_{ij}\right)}_{n\times n}$. In the light of Lemma 1, we can know that $c_i>0(i=1,2,\cdots ,n)$.

Meanwhile, by applying the combination identity from graph theory (see [14], Theorem 2.2) and utilizing the fact that $W\left(\mathcal{Q}\right)\ge 0$, taking the derivative of $V(\theta ,t)$ along the system trajectory of (4), we can conclude that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\mathrm{d}}^{\left(3\right)}V\left(\theta \left(t\right),t\right)}{\mathrm{d}t}}& \le & \sum _{i=1}^n{e}^{\eta t}{c}_i\left[-\left(2{\rho }_i-4\sum _{j=1}^n{b}_{ij}-{\sigma }_i-\eta \right){\theta }_i^2\left(t\right)\right]\hfill \\ & & +\sum _{i,j=1}^n{e}^{\eta t}{c}_i{b}_{ij}\left({\theta }_j^2\left(t\right)-{\theta }_i^2\left(t\right)\right)+\sum _{i=1}^n{e}^{\eta t}{c}_i{\sigma }_i{\xi }_i^2\left(t\right)\hfill \\ & \le & \sum _{i=1}^n{e}^{\eta t}{c}_i\left[-\left(2{\rho }_i-4\sum _{j=1}^n{b}_{ij}-{\sigma }_i-\eta \right){\theta }_i^2\left(t\right)\right]\hfill \\ & & +\sum _{i=1}^n{e}^{\eta t}{c}_i{\sigma }_i{\xi }_i^2\left(t\right)+\sum _{\mathcal{Q}\in \mathbb{Q}}W\left(\mathcal{Q}\right)\sum _{(j,i)\in \mathcal{E}\left({\mathcal{c}}_{\mathcal{Q}}\right)}{e}^{\eta t}\left({\theta }_j^2\left(t\right)-{\theta }_i^2\left(t\right)\right)\hfill \\ & \le & \sum _{i=1}^n{e}^{\eta t}{c}_i\left[-\left(2{\rho }_i-4\sum _{j=1}^n{b}_{ij}-{\sigma }_i-\eta \right){\theta }_i{\left(t\right)}^2+{\sigma }_i{\xi }_i^2\left(t\right)\right],\hfill \end{array}$$

where $\mathbb{Q}$ represents the set of all spanning unicyclic graphs of $(\mathcal{G},B)$, $W\left(\mathcal{Q}\right)$ is the weight of $\mathcal{Q}$, and ${\mathcal{C}}_{\mathcal{Q}}$ denotes the directed cycle within $\mathcal{Q}$. Differentiating $V(\theta ,t)$ along the trajectory of RCKOIC (1) gives

$$\begin{array}{ccc}{\displaystyle \frac{{\mathrm{d}}^{\left(1\right)}V\left(\theta \left(t\right),t\right)}{\mathrm{d}t}}& \le & {\displaystyle \frac{{\mathrm{d}}^{\left(3\right)}V\left(\theta \left(t\right),t\right)}{\mathrm{d}t}}+{\displaystyle \sum _{i,j=1}^n}{e}^{\eta t}{c}_i{\tau }_{ij}\left({\theta }_j{\left(t\right)}^2-{\theta }_i{\left(t\right)}^2\right)+3{\displaystyle \sum _{i,j=1}^n}{e}^{\eta t}{c}_i{\tau }_{ij}{\theta }_i{\left(t\right)}^2\hfill \\ & \le & {\displaystyle \sum _{i=1}^n}{e}^{\eta t}{c}_i\left[-\left(2{\rho }_i-{\displaystyle \sum _{j=1}^n}(4b_{ij}+3{\tau }_{ij})-{\sigma }_i-\eta \right){\theta }_i{\left(t\right)}^2+{\sigma }_i{\xi }_i{\left(t\right)}^2\right]\hfill \\ & \le & e^{\eta t}\left({\displaystyle \underset{1\le i\le n}{max}}\left\{c_i{\sigma }_i\right\}|\xi \left(t\right){|}^2-{\displaystyle \underset{1\le i\le n}{min}}\left\{c_i{\alpha }_i\right\}{\left|\theta \left(t\right)\right|}^2\right)\hfill \\ & \triangleq & -e^{\eta t}{\gamma }_2{\left|\theta \left(t\right)\right|}^2+e^{\eta t}{\gamma }_1{\left|\xi \left(t\right)\right|}^2,\hfill \end{array}$$

(5)

where ${\alpha }_i=2{\rho }_i-{\sum }_{j=1}^n(4b_{ij}+3{\tau }_{ij})-{\sigma }_i-\eta$, ${\gamma }_1=\underset{1\le i\le n}{max}\left\{c_i{\sigma }_i\right\}$, and ${\gamma }_2=\underset{1\le i\le n}{min}\left\{c_i{\alpha }_i\right\}.$ In line with condition (2) in Theorem 1, it is easy to derive ${\alpha }_i>0$. Taking the expectation of both sides of (5), in the light of Fubini’s theorem [15], we deduce

$$\begin{array}{c}\hfill \mathbb{E}{e}^{\eta t}\underline{\alpha }|\theta \left(t\right){|}^2\le V({\theta }_0,0)+{\int }_0^t\mathbb{E}\left[{\displaystyle \frac{{\mathrm{d}}^{\left(1\right)}V\left(\theta \left(s\right),s\right)}{\mathrm{d}s}}\right]\mathrm{d}s\le \overline{\alpha }|{\theta }_0{|}^2+{\gamma }_2\underset{0\le s\le t}{sup}\mathbb{E}{\left|\xi \left(s\right)\right|}^2{\int }_0^t{e}^{\eta s}\mathrm{d}s,\end{array}$$

(6)

where $\underline{\alpha }={\displaystyle \underset{1\le i\le n}{min}}\left\{c_i\right\}$ and $\overline{\alpha }={\displaystyle \underset{1\le i\le n}{max}}\left\{c_i\right\}$. Combining with (6) and Jensen’s inequality [16], it is easy to obtain

$$\begin{array}{ccc}\mathbb{E}\left|\theta \right(t\left)\right|& \le & {\left({\alpha }_1\right)}^{-1}\left(e^{-\eta t}{\alpha }_2\left(\right|{\theta }_0\left|\right)\right)+{\left({\alpha }_1\right)}^{-1}\left({\gamma }_2{\displaystyle \underset{0\le s\le t}{sup}}\mathbb{E}{\left|\xi \left(s\right)\right|}^2\right)\hfill \\ & \triangleq & \tilde{\beta }\left(\right|{\theta }_0|,t)+\tilde{\Gamma }\left({\displaystyle \underset{0\le s\le t}{sup}}\mathbb{E}{\left|\xi \left(s\right)\right|}^2\right),\hfill \end{array}$$

(7)

where ${\alpha }_1\left(\right|\theta \left|\right)=\underline{\alpha }{\left|\theta \right|}^2,$${\alpha }_2\left(\right|{\theta }_0\left|\right)=\overline{\alpha }{\left|{\theta }_0\right|}^2,\tilde{\Gamma }(·)\in {\mathcal{K}}_{\infty }$ and $\tilde{\beta }\left(\right|{\theta }_0|,t)\in {\mathcal{K}}_{\mathcal{L}}$. According to (7) and Chebychev’s inequality [17], for any $\epsilon >0$,

$$\begin{array}{c}\hfill \mathbb{P}\left\{\left|\theta \left(t\right)\right|>\beta \left(\right|{\theta }_0|,t)+\Gamma \left(\underset{0\le s\le t}{\sup }\mathbb{E}{\left|\xi \left(s\right)\right|}^2\right)\right\}\le {\displaystyle \frac{\mathbb{E}\left|\theta \right(t\left)\right|\epsilon }{\tilde{\beta }\left(\right|{\theta }_0|,t)+\tilde{\Gamma }\left(\underset{0\le s\le t}{\sup }\mathbb{E}{\left|\xi \left(s\right)\right|}^2\right)}}\le \epsilon ,\end{array}$$

where $\beta \left(\right|{\theta }_0|,t)={\displaystyle \frac{1}{\epsilon }}\tilde{\beta }\left(\right|{\theta }_0|,t)\in {\mathcal{K}}_{\mathcal{L}}$ and $\Gamma (·)={\displaystyle \frac{1}{\epsilon }}\tilde{\Gamma }(·)\in {\mathcal{K}}_{\infty }$. It is obvious that the inequality above is a complementary event of the condition in Definition 2, which means RCKOIC (1) is NSSP. Overall, the proof is accomplished. □

Remark 1. 

In Theorem 1 we give the sufficiency criterion for the RCKOIC (1) to satisfy the NSSP, and since its criterion is based on the uniqueness of the existence of the solution, in this paper we complete the proof of the consistent convergence of the iterative sequence and the basis for the establishment of the criterion by constructing the iterative sequence and using the Bellman–Gronwall inequality. It is easy to see that RCKOIC (1) is asymptotically stable through our designed control when the stochastic process ξ disappears from Definition 1. Furthermore, if the condition ${\displaystyle \underset{0\le s\le t}{sup}}\mathbb{E}{\left|\xi \left(s\right)\right|}^2<+\infty$ satisfied by the stochastic process ξ is changed to ${\displaystyle \underset{0\le s\le t}{sup}}\mathbb{E}{\left|\xi \left(s\right)\right|}^2<d{e}^{\lambda t}$, where d and λ are positive real numbers, the results in Theorem 1 still hold.

Remark 2. 

One of the main factors affecting the dynamics of RCKOIC (1) is its coupled topological structure, which is studied by many scholars, giving some essential conditions such as strong connectivity [18] or weak connectivity [2]. However, in many practical applications, the coupled topological structure of RCKOIC (1) does not have these conditions. In this paper, the coupled topological structure of RCKOIC (1) is arbitrary, which makes it more widely used in practice.

4. Numerical Test

We illustrate the practicality and applicability of the proposed theory through a numerical simulation example.

The number of oscillators in RCKOIC (1) is $n=4$. Next, we select the stochastic process ${\xi }_i\left(t\right)=icos(t+\varphi )$, where random variable $\varphi$ is a uniform distribution subject to $(0,2\pi )$. Obviously, we can deduce that ${\xi }_i$ is a wide stationary process and $\underset{0\le s\le t}{\sup }\mathbb{E}{\left|\xi \left(s\right)\right|}^2=15<+\infty .$

Let

$${\sigma }_1=2.3,{\sigma }_2=1.5,{\sigma }_3=1.2,{\sigma }_4=0.7,$$

and natural frequency

$${\omega }_1=0.27,{\omega }_2=0.99,{\omega }_3=1.48,{\omega }_4=2.66.$$

Further, the gain coefficient in the feedback control we designed are

$${\rho }_1=6.9,{\rho }_2=6.2,{\rho }_3=5.4,{\rho }_4=5.9.$$

The coupling configuration matrix

$$A=\left(\begin{array}{cccc}0& 0.98& 0.93& 0\\ 1.22& 0& 0& 0\\ 0& 0.11& 0& 0\\ 0& 0.99& 0& 0\end{array}\right).$$

The initial values are as follows:

$${\theta }_1\left(0\right)=0.1,{\theta }_2\left(0\right)=0.2,{\theta }_3\left(0\right)=0.3,{\theta }_4\left(0\right)=0.4.$$

For the numerical simulation, we applied an iterative step size of $h=0.01$ and a total simulation time of $T=6$ s. Clearly, the condition (2) in Theorem 1 is satisfied. Thus, RCKOIC (1) is NSSP, which is exhibited in Figure 1. To make readers see more clearly the effect of stochastic process $\xi$ on RCKOIC (1), as a comparison, the state trajectory of the solution for RCKOIC (1) without stochastic disturbance is given in Figure 2.

Through the theorem section of the article, we can see that the random disturbance intensity $\sigma$ and the gain coefficient $\rho$ in the feedback control are two important factors affecting the stability of RCKOIC (1). In order to make the reader more intuitively feel this, next we explore the effect of changing these two variables on RCKOIC (1), respectively.

Firstly, the intensity of the random perturbation ${\sigma }_i$ is reduced from the original randomly perturbed system, and variables such as gain coefficients and coupling relations are controlled to remain constant. Let

$${\sigma }_1=1.15,{\sigma }_2=0.75,{\sigma }_3=0.6,{\sigma }_4=0.35.$$

Figure 3 is obtained after going through the run.

At this point, condition (2) in Theorem 1 is also still satisfied, and RCKOIC (1) is NSSP. By comparing with Figure 1, it is obvious that the fluctuations of the state variables are relatively small and the trajectory is closer to a stable value or a smaller range of oscillations. This explains the fact that feedback control is more likely to maintain the stability of the system with smaller perturbations.

Subsequently, the gain coefficient ${\rho }_i$ in the feedback control of the system is increased based on the original stochastic perturbation system, and the variables such as the intensity of the stochastic perturbation and the coupling relationship are controlled to remain constant. Let

$${\rho }_1=13.8,{\rho }_2=12.4,{\rho }_3=10.8,{\rho }_4=11.8.$$

Figure 4 is obtained after going through the run.

At this point, RCKOIC (1) is NSSP. By comparing with Figure 1, a smoother trajectory appears, the range of fluctuation becomes smaller, and the system converges faster. This also indicates that the feedback control becomes stronger and the system is more likely to pull the deviated state back to the equilibrium position. In this case, even if there is a random perturbation, a higher ${\rho }_i$ is more effective in controlling the state variable to fluctuate within a smaller range.

5. Conclusions

In this paper, we design a feedback controller and construct appropriate Lyapunov functions for the RCKOIC (1) system based on Kirchhoff’s theorem in graph theory. By applying the Lyapunov method and stochastic techniques, the stochastic input-state stability (NSSP) of the RCKOIC (1) system under arbitrary coupling topology and second-order moments random perturbations is obtained. Finally, numerical examples and simulation results show that the feedback controller can effectively reduce the state fluctuations under random disturbances and make the system converge to equilibrium rapidly under disturbance-free conditions, which verifies the feasibility and validity of our theory. However, in real life, non-ideal connections between oscillators in the RCKOIC (1) system, finite signal propagation speeds, or other factors may lead to time delays, which affect the dynamic behavior of the RCKOIC (1) system. Therefore, it is important to consider the time delay in RCKOIC (1) systems, which is our future research direction.